1. ## Rolle's Theorem Problem

Let f be a twice differentiable function on R with f” continuous on [0, 1] such that
the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4).
Prove that there exists an xo in (0, 1) such that f”(xo) = 0.

Here is what I have so far....

the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4)
f(c1) (1-0) = 2 f(c2) (3/4 - 1/4)
for c1 belonging to (0, 1) and c2 belonging to (1/4, 3/4)
f(c1) = f(c2)
By Rolle's Theorem, there exists c3 that belongs to (c1, c2) such that f ' (c3)=0

I need another part similar to this so that the derivitive of these which would be f " (xo) = 0.

Any suggestions are welcome.

2. Many people here will refuse to read non-LaTeX math.

Originally Posted by page929
Let $f$ be a twice differentiable function on $\mathbb{R}$ with $f''$ continuous on $[0,1]$ such that $\int_0^1 f(x)\text{ }dx=2\int_{\frac{1}{4}}^{\frac{3}{4}}f(x)\text{ }dx$. Prove that there exists some $x_0\in(0,1)$ such that $f''(x_0)=0$
Is this the question?

3. Yes